DOCSLIB.ORG

On the Relation Between Yangians, Affine Hecke Algebras and Long-Range Interacting Models

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
On the Relation Between Yangians, Affine Hecke Algebras and Long-Range Interacting Models Recherche Coopérative sur Programme no 25 DENIS BERNARD On the Relation Between Yangians, Affine Hecke Algebras and Long-Range Interacting Models Les rencontres physiciens-mathématiciens de Strasbourg - RCP25, 1995, tome 47 « Conférences de M. Audin, D. Bernard, A. Bilal, B. Enriquez, E. Frenkel, F. Golse, M. Katz, R. Lawrence, O. Mathieu, P. Von Moerbeke, V. Ovsienko, N. Reshetikhin, S. Thei- sen », , exp. no 3, p. 77-91 <http://www.numdam.org/item?id=RCP25_1995__47__77_0> © Université Louis Pasteur (Strasbourg), 1995, tous droits réservés. L’accès aux archives de la série « Recherche Coopérative sur Programme no 25 » implique l’ac- cord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utili- sation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ On the relation between Yangians, affine Hecke algebras and long-range interacting models. Denis Bernard 2 Service de Physique Théorique de Saclay 3 F-91191, Gif-sur-Yvette, France. Plan : 1- Long-Range Interacting Models. 2- Yangians and Affine Hecke Algebras. 3- A Yangian deformation of the W-algebras. 1 Long-Range Interacting Models. There is a large family of integrable long range interacting spin chains. In particular, a model introduced by Haldane and Shastry [2], see also [3], is a variant of the spin half Heisenberg chain, with exchange inversely proportional to the square distance between the spins. It possesses the remarkable properties that its spectrum is highly degenerate and additive, and that the elementary excitations are spin half objects obeying a half-fractional statistics intermediate between bosons and fermions. They are defined as follows. We consider a spin chain with TV sites, labeled by integers z, j, · · · ranging from 1 to N. On each sites there is a spin variable σζ· which takes two values: G{ — ±. The hamiltonians, which are all su(2) invariant, are of the following form : Η = Σ h» (Ρ»-1) (1) where P{j is the operator which exchanges the spins at the sites i and j. For translation invariance h{j = h(i — j). Demanding the integrability of the model selects the functions h. 2Member of the CNRS 3Laboratoire de la Direction des Sciences de la Matière du Commisariat à l'Energie Atomique. 77 The possible choices are : {ύή£Ίχ)2, hyperbolic model (7 real) π h(x) = < / ^2, trigonometric model k Pf^), elliptic model. where V(x) is the Weierstrass function. When 7 —• 00, the hyperbolic model reduces to the Heisenberg spin chain: h{j = #t,j+i + and for 7 —» 0, the interaction becomes the 1/x2 exchange. The hyperbolic model has not been completly solved for general 7, although a partial list of eigenstates is known. The elliptic model is even more intriguing since it interpolates between the Heisenberg spin chain of finite length and the trigonometric model [4]· The Haldane-Shastry spin chain is the trigonometric model. In the thermodynamical limit, Ν —> oo, it reduces to the l/x2 exchange model, but it also possesses remarkable properties at finite TV. Notably, its hamiltonian commutes with an infinite dimensional algebra whose two first generators are [5] : Qo = Σ & (2) i Qi = Σ cot9 (^^Tr) & x $ (3) with Si the spin operators acting on the site i. The first generators are the usual su(2) generators. Together with the second ones, they form a representation of the su(2) Yangian, (which is a deformation of the su(2) current algebra, see section 3 for an introduction to the Yangians). This infinite dimensional symmetry is at the origin of the large degeneracy of the spectrum. The fact that the hamiltonian is Yangian invariant at finite ΛΓ is particular to the Haldane-Shastry spin chain; in the Heisenberg spin chain, the Yangian symmetry only appears in the thermodynamical limit. In order to grasp the rules describing the spectrum, we first construct few eigenstates. Clearly, the ferromagnetic vacuum |Ω) = | + Η + +) is an eigenstate : its energy is zero. The eigenstates in the one-magnon sector are the plane waves :\k) = Ση exp(i2Kkn/N)a~ |Ω), with pseudo-momentun fc, 1 < k < (Ν — 1): the one-magnon energy is e(k) — k(k — TV). In the two-magnon sectors, i.e. for states of the form \φ) = Ση,™ ^η,™0"~σ~ |Ω), the eigenstates which are not degenerate with the zero or one-magnon eigenstates are labeled by two pseudo-momenta &i, &2, with 1 < fci, k2 < (Ν — 1). They are given by : , ,71 I . ,771 ^[^1,^2] /L _ L\ (UJnki-]rmk2 . (jmk1^nk2\ ' fLJnk1+mk2 _ UJmk1+nk2\ with ω = exp(i2w/N). Note that these wave functions vanish if ki = k2 but also if \ki — k2\ = 1. The energy of \φ^Μ) is Ε = c(ifci) + e(k2). From the two-magnon computation we learn two properties of the spectrum : (i) it is additive, e.g. the two-magnon energy is the sum of the one-magnon energies, but (ii) the pseudo-momenta satisfy a selection rule : they are neither equal nor they differ by a unit. These rules are the general rules, and the full spectrum can be described as follows [6]. 78 To each eigenstate multiplet is associated a set of pseudo-momenta {kp} which are non- consecutive integers ranging from 1 to (TV — 1). The energy of an eigenstate |{&p}) with pseudo-momenta {kp} is: ' 7Γ N 2 H\{k }) = e(kp) \{kp}} with e(k) = k{k - TV) p (ς (4) Furthermore, the degeneracy of the multiplet with pseudo-momenta {kp} is described by its su(2) representation content as follows. Encode the pseudo-momenta in a sequence of (TV — 1) labels 0 or 1 in which the l's indicate the positions of the pseudo-momenta; add two O's at both extremities of the sequence which now has length (TV + 1). Since the pseudo- momenta are neither equal nor consecutive, two labels 1 cannot be adjacent. The sequence corresponding to the ferromagnetic vacuum is a line of 0, those of the one-magnon states have TV label 0 and only one label 1, and so on. A sequence can be decomposed into the product of elementary motifs, which are series of (Q + 1) consécutives O's. The multiplicity of the spectrum is recovered if to each elementray motif of length (Q + 1) we associate a spin Q/2 representation of su(2). The representation content of the full sequence is then given by the tensor product of its elementary motifs. The magnons are the excitations over the ferromagnetic vacuum; the excitations over the antiferromagnetic vacuum are conveniently described in terms of spinons. For Ν even, the antiferromagnetic vacuum corresponds to the alternating sequence of symbols 010101 · · · 010. The excitations are obtained by flipping and moving the symbols 0 and 1. Let us give the sequences corresponding to the first few excitations over the antiferromagnetic vacuum, (for concreteness we choose TV = 10) : ' 01010101010, antiferromagnetic vacuum (o) < 0 10 10 1 O^O^O 10, a two-spinon excitations (2a) w 0 1 0^0 1 0^0 10 10, a two-spinon excitations (2b), etc... We have inserted a χ between any two consecutive labels 0. These crosses represent the spinon excitations, their number is the spinon number. Note that there is no one-spinon excitation for TV even. By convention, we will say that consecutive crosses not separated by any label 1 correspond to spinons in the same orbital, while crosses separated by labels 1 correspond to spinons in different orbitals. From the rules described above, it follows that the degeneracy of the excitations (2a) and (2b) are different : it is three in the case (2a) and four in the case (2b). These degeneracy are recovered by giving a su(2) spin half to the spinons and by assuming that spinons in the same orbital are in a fully symmetric states. Hence, in the case (2a), there are two spinons in the same orbital and therefore they form a spin one representation of su(2), and in the case (2b), the two spinons are in two different orbitals and therefore they form a su(2) representation isomorphic to the tensor product of two spin half representations of su(2). The fact that the spinons are spin half excitations can also be seen by looking at the excitations of a spin chain of length TV with TV odd. This description of the states generalizes to the full spectrum. We can classify the se­ quences by their number M of pseudo-momenta. The spinon number Nsp of a sequence is sp then defined by M = ~2 . Since M is an integer, (N—Nsp) is always even : this means that the spinons are always created by pairs. A sequence of pseudo-momenta {kp; ρ =!,···, M}, 79 in the Nsp spinon sector, can be decomposed into (M +1) elementary motifs where, as before an elementary motif is a series of consecutive 0. We call the elementary motifs the accessible N sp orbitals to the spinons. At fixed Nsp, there are Norb = (l + ~^ ) orbitals. Hence, a sequence of pseudo-momenta {kp} corresponds to the filling of the Nori orbitals with respective spinon occupation numbers np = (kP+i — kp — 2), with k0 — — 1 and — Ν + 1 by convention. Since an elementary motif of length (Q + 1) corresponds to a spin Q/2 representation of su(2), the full degeneracy of the sequences is then recovered by assuming that the spinons are spin half objects which behave as bosons in each orbitals.
Load more

Recommended publications
  • Integrable Systems and Quantum Deformations
    Integrable Systems and Quantum Deformations Peter Koroteev University of Minnesota In collaboration with N. Beisert, F. Spill, A. Rej 0802.0777, work in progress MG12, July 13th 2009 Outline Introduction. Integrability and Symmetry What is Integrability Coordinate Bethe Ansatz AdS/CFT Correspondence Quantum Deformation of 1D Hubbard Model 2 Uq(su(2j2) n R ) algebra Hubbard-like Models Universal R-matrix Yangian Universal R-matrix I \Complicated", Chaotic models. Almost untractable I Integrable models: not necessrily easy/complicated Integrable models can be completely solved Why Integrability? I Easy models: Classical Mechanics (ossilator, free point particle) I Integrable models: not necessrily easy/complicated Integrable models can be completely solved Why Integrability? I Easy models: Classical Mechanics (ossilator, free point particle) I \Complicated", Chaotic models. Almost untractable Integrable models can be completely solved Why Integrability? I Easy models: Classical Mechanics (ossilator, free point particle) I \Complicated", Chaotic models. Almost untractable I Integrable models: not necessrily easy/complicated Why Integrability? I Easy models: Classical Mechanics (ossilator, free point particle) I \Complicated", Chaotic models. Almost untractable I Integrable models: not necessrily easy/complicated Integrable models can be completely solved I Finite dimensional: mechanical with n d.o.f.s { exist n-1 local integrals in involution I Field Theory: Infinite dimensional symmetry, S{matrix satisfies integrability constraints. We are
    [Show full text]
  • Yangians and Classical Lie Algebras Mathematical Surveys and Monographs Volume 143
    http://dx.doi.org/10.1090/surv/143 Yangians and Classical Lie Algebras Mathematical Surveys and Monographs Volume 143 Yangians and Classical Lie Algebras Alexander Molev American Mathematical Society ^VDED" EDITORIAL COMMITTEE Jerry L. Bona Michael G. Eastwood Ralph L. Cohen Michael P. Loss J. T. Stafford, Chair 2000 Mathematics Subject Classification. Primary 17B37, 81R50; Secondary 17B10, 17B20, 81R10, 05E10, 05E15. For additional information and updates on this book, visit www.ams.org/bookpages/SURV-143 Library of Congress Cataloging-in-Publication Data Molev, Alexander, 1961- Yangians and classical Lie algebras / Alexander Molev. p. cm. — (Mathematical surveys and monographs, ISSN 0076-5376 ; v. 143) Includes bibliographical references and index. ISBN 978-0-8218-4374-1 (alk. paper) 1. Lie algebras. 2. Quantum groups. 3. Matrices. 4. Representations of algebras. 5. Symmetry (Mathematics). I. Title. QA252.3.M65 2007 512/.482—dc22 2007060781 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to [email protected]. © 2007 by the American Mathematical Society.
    [Show full text]
  • An Introduction to Yangian Symmetries L
    - SPhT-92-134 An Introduction to Yangian Symmetries l Denis Bernard Service de Physique Théorique de Saclay 2 F-91191, Gif-sur- Yvette, France. We review some aspects of the quantum Yangians as symmetry algebras of two-dimensional quantum field theories. The plan of these notes is the following: 1 - The classical Heisenberg model: •Non-Abelian symmetries; «The generators of the symmetries and the semi-classical Yangians; «An alternative presentation of the semi-classical Yangians; «Digression on Poisson-Lie groups. 2 - The quantum Heisenberg chain: •Non-Abelian symmetries and the quantum Yangians; «The transfer matrix and an alternative pre- sentation of the Yangians; «Digression on the double Yangians. Quantum integrable models are characterized by the existence of commuting conserved charges which one is free to choose as Hamiltonians. They usually also possess extra symmetries which are at the origin of degeneracies in the spectrum. These symmetries are often not a group but a quantum group. The Yangians [1] are quantum group symmetries which (almost systematically) show up in integrable models invariant un- der an internal non-Abelian group. The most standard example of this kind is the Heisenberg model which we use in these notes for presenting an introduction to the Yangian algebras. Other examples of two-dimensional relativistic quantum field theo- ries which are Yangian invariant are provided e.g. by sigma models, two dimensional massive current algebras, principal chiral models with or without topological terms [2, 3], etc... Usually the infinite dimensional quantum symmetries are not compatible with periodic boundary conditions. However more recently, it has been realized that a variant of the Heisenberg chain with a long-range interaction was Yangian invariant in a way compatible with periodic boundary conditions, making eigenstates countable and the degeneracy easy to describe [4].
    [Show full text]
  • Yangian Realisations from Finite W-Algebras
    LAPTH Laboratoire d’Annecy-le-Vieux de Physique Th´eorique Yangian realisations from finite W-algebras E. Ragoucya,1 and P. Sorbab,2 a Theory Division, CERN, CH-1211 Gen`eve 23 b Laboratoire de Physique Th´eorique LAPTH3 LAPP, BP 110, F-74941 Annecy-le-Vieux Cedex, France. Abstract We construct an algebra homomorphism between the Yangian Y (sl(n)) and the finite W-algebras W(sl(np),n.sl(p)) for any p. We show how this result can be applied to determine properties of the finite dimensional representations of such W-algebras. hep-th/9803243 CERN-TH/98-104 LAPTH-672/98 March 1998 [email protected] On leave of absence from LAPTH. [email protected]. 3URA 14-36 du CNRS, associ´ee `a l’Universit´edeSavoie. Contents 1 Introduction 2 2 Finite W algebras: notation and classification. 3 2.1 Classical W(G, S)algebras.............................. 3 2.2 Quantum W(G, S)algebras............................. 4 2.3Miurarepresentations................................. 5 2.4 Example: W(sl(np),n.sl(p)))algebras....................... 5 3 Yangians Y (G) 7 3.1Definition....................................... 7 3.2 Evaluation representations of Y (sl(n))....................... 8 4 Yangians and classical W-algebra 8 5 Yangians and quantum W(sl(np),n.sl(p)) algebras 9 6 Representations of W(sl(2n),n.sl(2)) algebras 11 7 Conclusion 13 A The soldering procedure 14 B Classical W(sl(np),n.sl(p)) algebras 16 B.1Generalities...................................... 16 B.2 The generic case n 6=2................................ 18 B.3 The particular case of Y (sl(2))..........................
    [Show full text]
  • The RTT-Realization for the Deformed Gl(2|2) Yangian
    arxiv:1401.7691 The RTT-Realization for the Deformed gl(2j2) Yangian Niklas Beisert and Marius de Leeuw Institut f¨urTheoretische Physik, Eidgen¨ossischeTechnische Hochschule Z¨urich Wolfgang-Pauli-Strasse 27, 8093 Z¨urich,Switzerland {nbeisert,deleeuwm}@itp.phys.ethz.ch Abstract In this paper we work out the RTT-realization for the Yangian algebra of the Hubbard model and AdS/CFT correspondence. We find that this Yangian algebra is of a non-standard type in which the levels of the Yangian mix. The crucial feature that allows this is a braiding factor that deforms the coproduct and generates the central extensions of the underlying sl(2j2) Lie algebra. In our RTT-realization we have arXiv:1401.7691v2 [math-ph] 23 Mar 2018 also been able to incorporate the so-called secret symmetry and we were able to extend it to higher Yangian levels. Finally, we discuss the center of the Yangian and the automorphisms related to crossing symmetry. 1 Introduction Integrable systems constitute a class of models in physics and mathematics that are, more or less, characterized by the fact that they, in some sense, can be solved exactly. They describe simple but rich models of physical phenomena like magnetism (the Heisenberg spin chain) and superconductivity (the Hubbard model). Their exact solvability is con- nected to the fact that there is an intimate relation between integrable models and infinite dimensional algebras which is one of the reasons integrable systems have found their way into mathematics. In a sense, many integrable models can be classified by their (infinite) symmetry algebra and its representation theory.
    [Show full text]
  • Hopf Algebras, Quantum Groups and Yang-Baxter Equations
    axioms Hopf Algebras, Quantum Groups and Yang-Baxter Equations Edited by Florin Felix Nichita Printed Edition of the Special Issue Published in Axioms www.mdpi.com/journal/axioms Hopf Algebras, Quantum Groups and Yang-Baxter Equations Hopf Algebras, Quantum Groups and Yang-Baxter Equations Special Issue Editor Florin Felix Nichita MDPI • Basel • Beijing • Wuhan • Barcelona • Belgrade Special Issue Editor Florin Felix Nichita Simion Stoilow Institute of Mathematics of the Romanian Academy Romania Editorial Office MDPI St. Alban-Anlage 66 4052 Basel, Switzerland This is a reprint of articles from the Special Issue published online in the open access journal Axioms (ISSN 2075-1680) in 2012 (available at: https://www.mdpi.com/journal/axioms/special issues/ hopf algebras) For citation purposes, cite each article independently as indicated on the article page online and as indicated below: LastName, A.A.; LastName, B.B.; LastName, C.C. Article Title. Journal Name Year, Article Number, Page Range. ISBN 978-3-03897-324-9 (Pbk) ISBN 978-3-03897-325-6 (PDF) c 2019 by the authors. Articles in this book are Open Access and distributed under the Creative Commons Attribution (CC BY) license, which allows users to download, copy and build upon published articles, as long as the author and publisher are properly credited, which ensures maximum dissemination and a wider impact of our publications. The book as a whole is distributed by MDPI under the terms and conditions of the Creative Commons license CC BY-NC-ND. Contents About the Special Issue Editor ...................................... vii Preface to ”Hopf Algebras, Quantum Groups and Yang-Baxter Equations” ...........
    [Show full text]
  • Arxiv:1512.07178V2 [Hep-Th] 17 Jul 2019
    W-symmetry, topological vertex and affine Yangian Tom´aˇsProch´azka1 Arnold Sommerfeld Center for Theoretical Physics Ludwig Maximilian University of Munich Theresienstr. 37, D-80333 M¨unchen,Germany Institute of Physics AS CR Na Slovance 2, Prague 8, Czech Republic Abstract We discuss the representation theory of the non-linear chiral algebra W1+1 of Gaberdiel and Gopakumar and its connection to the Yangian of ud(1) whose presentation was given by Tsymbaliuk. The characters of completely degenerate representations of W1+1 are given by the topological vertex. The Yangian picture provides an infinite number of commut- ing charges which can be explicitly diagonalized in W1+1 highest weight representations. Many properties that are difficult to study in the W1+1 picture turn out to have a simple combinatorial interpretation, once translated to the Yangian picture. arXiv:1512.07178v2 [hep-th] 17 Jul 2019 1Email: tomas.prochazka at lmu.de 1 Contents 1 Introduction 3 2 Affine Yangian of gl(1) 5 2.1 Presentation . .6 2.2 Basic properties . .9 2.3 Spectral shift . 10 2.4 Affine u(1) and Virasoro subalgebras . 11 2.5 Coproduct . 15 3 Linear W1+1 and Y 16 3.1 Differential operators on circle . 16 3.2 Affine u(1) and Virasoro subalgebras . 17 3.3 Pope-Shen-Romans basis . 18 3.4 Linear W1+1 and Y ................................ 19 3.5 Free fermion representation . 21 3.6 Free boson representation . 22 4 Representations of Y 24 4.1 Highest weight representations . 24 4.2 Quasi-finite representations . 25 4.3 Vacuum MacMahon representation .
    [Show full text]
  • Twenty-Five Years of Representation Theory of Quantum Groups
    Twenty-five years of representation theory of quantum groups Pavel Etingof (MIT), Victor Ginzburg (University of Chicago), Nicolas Guay (University of Alberta), David Hernandez (Universite´ de Paris 7), Alistair Savage (University of Ottawa) August 7, 2011 – August 12, 2011 1 Overview of the Field Quantum groups first appeared under different disguises in the first half of the 1980’s, in particular in the work of physicists and mathematicians interested in the quantum inverse scattering method in statistical mechanics. For instance, one of the simplest examples, the quantized enveloping algebra of sl2, first surfaced in a paper of P. Kulish and N. Reshetikhin in 1981 about integrable systems. Quantized enveloping algebras of Kac-Moody algebras, the most studied examples of quantum groups, are non-commutative and non-cocommutative Hopf algebras discovered around 1985 independently by V. Drinfeld and M. Jimbo. The address of V. Drinfeld at the International Congress of Mathematicians in 1986 brought quantum groups to the attention of mathe- maticians worldwide and laid the foundations of this theory. In one fell swoop, Drinfeld gave birth to a new branch of algebra which started growing at tremendous speed and has remained a very active area of research to this day. Quantum groups have found applications in other branches of mathematics, topology being one notable example present at the workshop. Quantum groups have even made their appearance in subjects more distant from algebra, for instance in probability theory, harmonic analysis and number theory (the last one being the subject of the BIRS workshop “Whittaker functions, crystal bases and quantum groups” held in June 2010).
    [Show full text]
  • Yangians and Their Applications
    Yangians and their applications∗ A. I. Molev School of Mathematics and Statistics University of Sydney, NSW 2006, Australia [email protected] arXiv:math/0211288v1 [math.QA] 19 Nov 2002 ∗To appear in Handbook of Algebra (M. Hazewinkel, Ed.), Vol. 3, Elsevier. 1 Contents 1 Introduction 1 2 The Yangian for the general linear Lie algebra 3 2.1 Algebraic motivations and definitions . 3 2.2 Amatrixformofthedefiningrelations . 5 2.3 Automorphismsandanti-automorphisms . 6 2.4 Poincar´e–Birkhoff–Witttheorem. 7 2.5 Hopfalgebrastructure .......................... 8 2.6 Quantumdeterminantandquantumminors . 9 2.7 The center of Y(n) ............................ 11 2.8 The Yangian for the special linear Lie algebra . 12 2.9 Two more realizations of the Hopf algebra Y(sl2) ........... 13 2.10 Quantum Liouville formula . 15 2.11 Factorizationofthequantumdeterminant . 17 2.12 QuantumSylvestertheorem . 18 2.13 Thecentralizerconstruction . 19 2.14 Commutativesubalgebras . 21 3 The twisted Yangians 24 3.1 Definingrelations............................. 24 3.2 EmbeddingintotheYangian. 26 3.3 Sklyanindeterminant........................... 27 3.4 ThecenterofthetwistedYangian. 29 3.5 ThespecialtwistedYangian . 30 3.6 The quantum Liouville formula . 30 3.7 Factorization of the Sklyanin determinant . 31 3.8 Thecentralizerconstruction . 32 3.9 Commutativesubalgebras . 34 4 Applications to classical Lie algebras 35 4.1 Newton’sformulas ............................ 36 4.2 Cayley–Hamiltontheorem . .. .. 37 4.3 Graphical constructions of Casimir elements . 39 4.4 PfaffiansandHafnians .......................... 41 2 1 Introduction The term Yangian was introduced by Drinfeld (in honor of C. N. Yang) in his fun- damental paper [35] (1985). In that paper, besides the Yangians, Drinfeld defined the quantized Kac–Moody algebras which together with the work of Jimbo [64], who introduced these algebras independently, marked the beginning of the era of quantum groups.
    [Show full text]
  • Lectures on Yangian Symmetry Florian Loebbert
    HU-EP-16/12 Lectures on Yangian Symmetry Florian Loebbert Institut für Physik and IRIS Adlershof, Humboldt-Universität zu Berlin, Zum Großen Windkanal 6, D-12489 Berlin, Germany [email protected] Abstract In these introductory lectures we discuss the topic of Yangian symmetry from various perspectives. Forming the classical counterpart of the Yangian and an extension of ordinary Noether symmetries, first the concept of nonlocal charges in classical, two-dimensional field theory is reviewed. We then define the Yangian algebra following Drinfel’d’s original motivation to construct solutions to the quantum Yang–Baxter equation. Different realizations of the Yangian and its mathematical role as a Hopf algebra and quantum group are discussed. We demonstrate how the Yangian algebra is implemented in quantum, two-dimensional field theories and how its generators are renormalized. Implications of Yangian symmetry on the two- dimensional scattering matrix are investigated. We furthermore consider the important case of discrete Yangian symmetry realized on integrable spin chains. Finally we give a brief introduction to Yangian symmetry in planar, four-dimensional super Yang–Mills theory and indicate its impact on the dilatation operator and tree-level scattering amplitudes. These lectures are arXiv:1606.02947v2 [hep-th] 26 Jul 2016 illustrated by several examples, in particular the two-dimensional chiral Gross–Neveu model, the Heisenberg spin chain and N = 4 superconformal Yang–Mills theory in four dimensions. This review arose from lectures given at the Young Researchers Integrability School at Durham University (UK). Contents 1 Introduction 2 2 Classical Integrability and Non-local Charges in 2d Field Theory 4 2.1 Local and Bilocal Symmetries .
    [Show full text]